Title1: On global dynamics of three dimensional magnetohydrodynamics: nonlinear stability of Alfvén waves
Speaker: Pro.Lingbing He(Tsinghua University)
Time: Fri, Dec.14, 2018 , PM:15:30-16:30
Location: Room 4318, Building No.4, Wushan Campus
Abstract：
We construct and study global solutions for the 3-dimensional incompressible MHD systems with arbitrary small viscosity. In particular, we provide a rigorous justification for the following dynamical phenomenon observed in many contexts: the solution at the beginning behave like non-dispersive waves and the shape of the solution persists for a very long time (proportional to the Reynolds number); thereafter, the solution will be damped due to the long-time accumulation of the diffusive effects; eventually, the total energy of the system becomes extremely small compared to the viscosity so that the diffusion takes over and the solution afterwards decays fast in time. We do not assume any symmetry condition. The size of data and the a priori estimates do not depend on viscosity. The proof is builded upon a novel use of the basic energy identity and a geometric study of the characteristic hypersurfaces. The approach is partly inspired by Christodoulou-Klainerman’s proof of the nonlinear stability of Minkowski space in general relativity. This is a joint work with Li XU (Chinese Academy of Sciences) and Pin YU (Tsinghua University).
Title2: Global-in-time stability of large solutions for compressible viscous and heat-conductive gases
Speaker: Mr.Chao Wang(Peking University)
Time: Fri, Dec.14, 2018 , PM:16:30-17:30
Location: Room 4318, Building No.4, Wushan Campus
Abstract：
The talk is devoted to the investigation of the global-in-time stability of large solutions for the full Navier-Stokes-Fourier system in the whole space. Suppose that the density and the temperature are bounded from above uniformly in time in the Holder space $C^\alpha$ with $\alpha$ sufficiently small and in $L^\infty$ space respectively. Then we prove two results:
(1). Such kind of the solution will converge to its associated equilibrium with a rate which is the same as that for the heat equation if we impose the same condition on the initial data. As a result, we obtain the propagation of positive lower bounds of the density and the temperature.
(2). Such kind of the solution is stable, that is, any perturbed solution will remain close to the reference solution if initially they are close to each other. This shows that the set of the smooth and bounded solutions is open. This work joints with L. He and J. Huang.
